The Bogoiubov-de Gennes equations are used for a number of
theoretical
works on the trapped Bose-Einsetein condensates. Particularly,
it is
important that if all of the eigenvalues of the equations are
real, the
solutions of the equations diagonalize the unperturbed
Hamiltonian, and
the quasi-particle picture, which describes the quantum
fluctuation
around
the condensates, is obtained. We consider the quantum
fluctuation in the case that these equations have complex
eigenvalues.
First, to expand quantum field which represents the quantum
fluctuation,
we give the complete set including pairs of complex modes whose
eigenvalues are complex conjugate to each other. The expansion
of the
quantum field brings the operators associated with the complex
modes,
which are simply neither bosonic nor fermionic ones. Next, to
evaluate
physical quantities, we construct the eigenstate of the complex
mode
sector of the unperturbed Hamiltonian. Finally, we discuss the
instability of the condensates caused by the quantum fluctuation
associated with the complex mode in the context of Kubo's linear
response theory.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2007.MAR.W32.10